P-subnormal subgroups and the structure of finite groups

LetG be a finite group. Asubgroup H ofG is called aP-subnormal subgroup whenever H = G or there exists a chain of subgroups H = H-0 <= H-1 <= center dot center dot center dot <= H-t = G such that vertical bar H-i : Hi-1 vertical bar is a prime for every i is an element of {1, 2, ... t}. In the present paper, we study finite groups in which some 2-maximal subgroups or cyclic subgroups are P-subnormal subgroups. Several conditions for G to possess an ordered Sylow tower of supersolvable type are given.